cp's OEIS Frontend

A182486 Expansion of 2 * (2 + x) / (4 - x - x * sqrt(1 - 4*x)) in powers of x.

%I A182486 #23 Sep 08 2022 08:45:55
%S A182486 1,1,0,-1,-2,-4,-10,-29,-90,-290,-960,-3246,-11164,-38934,-137358,
%T A182486 -489341,-1757882,-6360634,-23160528,-84802606,-312041692,-1153271984,
%U A182486 -4279311348,-15935808866,-59537435012,-223099337404,-838282693560,-3157706225584,-11922241414880
%N A182486 Expansion of 2 * (2 + x) / (4 - x - x * sqrt(1 - 4*x)) in powers of x.
%C A182486 HANKEL transform of sequence and the sequence omitting a(0) is the sequence A033999(n) = (-1)^n. This is the unique sequence with that property.
%H A182486 G. C. Greubel, <a href="/A182486/b182486.txt">Table of n, a(n) for n = 0..1000</a>
%H A182486 Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%F A182486 G.f.: 2 * (2 + x) / (4 - x - x * sqrt(1 - 4*x)).
%F A182486 G.f.: (4 - x + x * sqrt(1 - 4*x)) / (2 * (2 - 2*x + x^2)).
%F A182486 G.f.: 1 / (1 - x / (1 + x / (1 - x / (1 - x / (1 - x / ...))))).
%F A182486 a(n) = -A135334(n - 2) if n > 2. a(n) - a(n+1) = A035929(n).
%F A182486 D-finite with recurrence: 2*(-n+1)*a(n) +2*(5*n-11)*a(n-1) +3*(-3*n+7)*a(n-2) +2*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Jun 08 2016
%e A182486 G.f. = 1 + x - x^3 - 2*x^4 - 4*x^5 - 10*x^6 - 29*x^7 - 90*x^8 - 290*x^9 + ...
%e A182486 1 = det([ 1]) = det([ 1]). -1 = det([ 1, 1; 1, 0]) = det([ 1, 0; 0, -1]). 1 = det([ 1, 1, 0; 1, 0, -1; 0, -1, -2]) = det([ 1, 0, -1; 0, -1, -2; -2, -4, -10]).
%t A182486 CoefficientList[Series[2*(2+x)/(4-x -x*Sqrt[1-4*x]), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 11 2018 *)
%o A182486 (PARI) {a(n) = if( n<0, 0, polcoeff( 2 * (2 + x) / (4 - x - x * sqrt(1 - 4*x + x * O(x^n))) ,n))}
%o A182486 (PARI) {a(n) = if( n<1, n==0, polcoeff( subst( (1 + x) / (1 + x^2), x, serreverse( x - x^2 + x * O(x^n))), n))}
%o A182486 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(2*(2+x)/(4-x -x*Sqrt(1-4*x)))); // _G. C. Greubel_, Aug 11 2018
%Y A182486 Cf. A033999, A035929, A135334.
%K A182486 sign
%O A182486 0,5
%A A182486 _Michael Somos_, May 02 2012